Lilyana runs a cake decorating business, for which $10\%$ of her orders come over the telephone. Let $C$ be the number of cake orders Lilyana receives in a month until she first gets an order over the telephone. Assume the method of placing each cake order is independent. Find the probability that it takes fewer than $5$ orders for Lilyana to get her first telephone order of the month. You may round your answer to the nearest hundredth. $P(C<5)=$
Without a fancy calculator On each order: $P({\text{telephone}})=0.1$ $P(\text{other}})=0.9$ If it takes fewer than $5$ orders for Lilyana to get her first phone order of the month, here are the possible sequences of orders: phone other, phone other, other, phone other, other, other, phone We could find the probability of each sequence and add those probabilities together. However, it would be faster to take the complement of the probability that none of the first $4$ orders is a phone order. $\begin{aligned} P(C<5) &= 1-P(4\text{ ordered other ways})\\\\ &=1-(0.9})^4 \\\\ &=1-0.6561\\\\ &=0.3439 \end{aligned}$ [Is there another way?] $P(C<5)=0.3439$